HARMONIC UNIVALENT FUNCTIONS INVOLVING
FOX - WRIGHT
SAIBAH SIREGAR, NORHEZAN UMAR ANDTN. AZMAR TN.DAUD
Abstract.In this paper we introduce the new subclass of analytic and harmonic univalent func-tions involving Fox-Wright funcfunc-tions.The coefficient bounds and Growth Theorem as well as Dis-tortion Theorem results for these functions are obtained.
1. INTRODUCTION
A continuous functionf(u,v) =u+ivis a complex-valued harmonic function
in a simply connected complex domain Cif both u and v are real harmonic in
C. LetH denote the family of function f =h+gwhich are harmonic and sense
-preserving in the open unit diskU=
z:|z|<1 wherehandgare given by
h(z) =z+ ∞
∑
k=2akzk , g(z) = ∞
∑
k=1bkzk (1)
The functionhis called the analytic part and g is called the co-analytic part
of the harmonic function f=h+g. The classHreduces to the class of normalized
univalent analytic functions, if the co-analytic part of f is zero. A necessary and
sufficient condition for f inH to be locally univalent and sense-preserving inU
is thath′(z) >
g
′(z)
inU . LetT H be the class of functions inH that may be
expressed as f=h+gwhere Received 06-12-2016, Accepted 26-12-2016. 2010 Mathematics Subject Classification: 30C45
Key words and Phrases: Harmonic, Univalent, starlike, Fox-Wright functions.
h(z) =z− ∞
∑
k=2akzk , g(z) = ∞
∑
k=1bkzk ,ak≥0,bk≥0 (2)
LetSH(α)andKH(α)be the subclasses ofHconsisting of univalent harmonic
functions starlike of order and convex of orderα, respectively, where0≤α<1
and|z|=r<1, if ∂
∂ θ(arg f(re
iθ))≥α
, |z|=r<1, (3)
and
∂ ∂ θ
" arg
∂ ∂ θ f(re
iθ)
#
≥α, |z|=r<1, (4)
Also letT SH(α)andT KH(α) be the respective subclasses ofSH(α)andKH(α) consisting of functions of the form (2).
Harmonic functions are indeed famous for their use in the study of minimal sur-faces and play an important role in a variety of problems in applied mathematics.
Harmonic functions have been studied by different geometers such as Kneser [8].
In 1984, Clunie and Sheil-Small [2] began a study of complex-valued, harmonic mappings defined on a domainU⊂C.
The coefficients bounds for the classesSH(α),KH(α),T SH(α),T KH(α)a are
stud-ied in details by Silverman [11], Jahangiri [4], Jahangiri & Silverman [5], Silverman
& Silvia [12],Jakubowski et al [6], Janteng et. al [7], Yalcin, [13], [14], [15] and Siregar et.al [10].
2. PRELIMINARY RESULTS In this paper, we considered the function as follows
Imα,k(α1,β1)f(z) =z+ ∞
∑
k=2Ωm
kΘmkakzk, (5)
Ωmk =
q
∏
j=1Γ αj+Aj(k−1)
s
∏
j=1Γ βj+Bj(k−1)
1 (k−1)!
, (6)
Θmk = "
(k−1)k(k+λ−2)! λ!(k−2)!
#
In this section, the classesSH(Ω,Θ,γ)andTH(Ω,Θ,γ), of functions which
are harmonic univalent inUwill be introduced. Some properties for functions f
belonging to these classes which include the coefficient estimates, growth results and distortion theorem will be given.
Let f =h+gdenote in the form (2). f be in class function ofSH(Ω,Θ,γ)if satisfy the condition
Re
3. COEFFICIENT BOUNDS Theorem 1.Let f =h+g with h and g given by (2),
By using triangle inequality for the above function is,
using the properties of sigma, from equation (11), we obtain
equation (6) and (7) respectively.
f(z1)−f(z2)
≥ |z1−z2|(1−|b1| −|z2||γ| −|b1|)>0
Consequently, f is univalent inU. To prove that f is sense preserving inU. This is
because
NextImλ,k(α1,β1)f(z)in equation (5),
The harmonic mappings
f(z) =z+
The restriction placed in Theorem 1 on the moduli of the coefficients of f=h+g
enables us to conclude for arbitrary rotation of the coefficients of fthat the resulting
Next the condition (10) is also necessary for functions f to be inTH(Ω,Θ,γ). Theorem 2.Let f =h+gwithhandggiven by (2). Then f ∈TH(Ω,Θ,γ)if and
only if the inequality (9) holds for the coefficients of f =h+g.
Proof. First suppose that f ∈TH(Ω,Θ,γ), then by (8) have Re
n
(Imλ,kh(z))′−Imλ,kg(z)o
=Re (
1− ∞
∑
k=2k[Ωm
kΘmk]akzk−1− ∞
∑
k=1k[Ωm
kΘmk]bkzk−1 )
>1−|γ|
If choosezto be real and letz→1−, then we can have1− ∞
∑
k=2k[Ωm
kΘmk]|ak| − ∞
∑
k=1k[Ωm
kΘmk]|bk|>1−|γ|, which is precisely the assertion (9). Conversely, suppose that the inequality (9) holds true. Then can be find from the equation (8) that
Re n
(Imλ,kh(z))′−Imλ,kg(z)o
=Re (
1− ∞
∑
k=2k[ΩmkΘmk]akzk−1− ∞
∑
k=1k[ΩmkΘmk]bkzk−1 )
≥2− ∞
∑
k=1k[ΩmkΘmk](|ak|+|bk|)|z|k−1
>2− ∞
∑
k=1k[ΩmkΘmk](|ak|+|bk|)≥1−|γ| provided that the inequality (9) is satisfied.
4. GROWTH BOUNDS AND DISTORTION THEOREM
In this subsection, growth bounds for functions inTH(Ω,Θ,γ)will be obtained and extreme points for this class will be given.
Theorem 3.If f ∈TH(Ω,Θ,γ)for0<|γ| ≤1,N0,λ≤0and|z|=r>1, then
f(z)
≤(1+|b1|)r+
|γ| −|b1| 2[Ωm
kΘmk] r2
and
f(z)
≥(1−|b1|)r−
|γ| −|b1| 2[Ωm
Proof. Let f∈TH(Ω,Θ,γ). Taking the absolute value of f(z). The right hand side
f(z)
≥ (1+|b1|)r− ∞
∑
k=2[|ak|+|bk|]rk
≥ (1+|b1|)r− ∞
∑
k=2[|ak|+|bk|]r2
≥(1+|b1|)r−|γ| −|b1| 2[Ωm
kΘmk] ∞
∑
k=22[Ωm kΘmk]
|γ| −|b1|[|ak|+|bk|]r 2
By equation (9), we obtain
f(z)
≥(1+|b1|)r−
|γ| −|b1| 2[Ωm
kΘmk]
r2 (14)
The left hand side,
f(z)
≤ (1+|b1|)r+ ∞
∑
k=2[|ak|+|bk|]rk
≤ (1+|b1|)r+ ∞
∑
k=2[|ak|+|bk|]r2
≤(1+|b1|)r+
|γ| −|b1| 2[Ωm
kΘmk] r2
∞
∑
k=22[Ωm kΘmk]
|γ| −|b1|[|ak|+|bk|]r 2
And also, by equation (9), we find
f(z)
≤(1+|b1|)r+
|γ| −|b1| 2[Ωm
kΘmk]
r2 (15)
The proof is complete.
Distortion for function inTH(Ω,Θ,γ)will be obtained by Theorem 4.
Theorem 4.If f ∈TH(Ω,Θ,γ)for0<|γ| ≤1, m∈N0,λ≥0and|z|=r>1
f′(z)
≤(1+|b1|) +
|γ| −|b1|
Ωm 2Θm2
r,
and
f′(z)
≥(1−|b1|)−
|γ| −|b1|
Ωm 2Θm2
To obtained distortion theorem, it can be differentiate the equation in (14) and
(15), then
f′(z)
≤(1+|b1|) +
|γ| −|b1|
Ωm 2Θm2
r,
and
f′(z)
≥(1−|b1|)−
|γ| −|b1|
Ωm 2Θm2
r.
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Saibah Siregar: Department of Science and Biotechnology, Faculty of Engineering and
Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia
E-mail: [email protected]
Norhezan Umar: Department of Science and Biotechnology, Faculty of Engineering and
Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia
E-mail: [email protected]
Tn. Azmar Tn.Daud: Department of Science and Biotechnology, Faculty of Engineering
and Life Sciences, University of Selangor, Bestari Jaya 45600, Selangor D.E. Malaysia
